Parameter Identification of Lorenz System with Incomplete Information: Case of One Unknown Function

Abstract

Inverse problem of the Lorenz system parametric identification is considered in the case of incomplete information about solutions of the system. In the present paper, it is assumed that only two solutions of the system from three are known in different combinations. The problem of the parameter identification of the system is solved by means of elimination of unknown functions from the original system. The obtained system of equations has the same order as the original one, but contains the unknown original parameters in new combinations. Sometimes, the number of new unknown parameters is higher than number of the original unknowns. In this case, the method of the constrained least squares minimization is used in the special formulation, developed by the authors. This novel formulation exploits linearity of the system with respect to the new unknown parameters, by means of which the number of nonlinear equations becomes equal to the number of the constraints between the new parameters. Two methods of the constraint minimization are considered: the classical method of Lagrange’s multipliers and a novel method of the auxiliary parameters. Numerical simulations demonstrate effectiveness of the algorithms.

Parameter Identification of Lorenz System with Incomplete Information: Case of One Known and Two Unknown Functions

“…Like in the previous paper let us consider the Lorenz system of equations: where a, b, c, d, f, g are unknown scalar parameters, which must be identified by means of solution of inverse problem with incomplete information. In [1], we assumed that two of three functions x(t), y(t), z(t) are known. In the present paper we assume that only one of three functions is known at initial time interval t ∈ [0, T ] in N + 1 points, corresponding to equidistant time instants, t i = T N i, where i = 0, 1, .…”

“…Substituting (1) in the second equation of system (1) and solving it with respect to z = z(t) we obtain:…”

“…In the previous paper of this series (see [1]) we considered the problem of parameter identification of the Lorenz system of equations assuming that one of three functions is unknown and other two functions are known in discrete equidistant time instants in an initial time interval. Now we assume that discrete values of only one function are known in the above-mentioned time instants and other two functions are unknown.…”

“…Realizing the approach developed in this present paper, we divide the problem into two parts. In the first part, we find the guess values of the unknown parameters using two methods which were introduced in the first paper (see [1]), i.e. either by the Lagrange multiplier method or by the method of auxiliary parameters.…”

“…Possible fields of application of the developed methods are outlined in the references . See [1] for a comprehensive background (as well as the motive) of the research problem described in this paper.…”